Proof of the Riemann Hypothesis utilizing the theory of Alternative Facts

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Description: Conway's powerful theory of Alternative Facts can render many difficult problems
tractable. Here we demonstrate the power of AF to prove the Riemann Hypothesis,
one of the most important unsolved...

Conway's powerful theory of Alternative Facts can render many difficult problems
tractable. Here we demonstrate the power of AF to prove the Riemann Hypothesis,
one of the most important unsolved problems in mathematics. We further suggest
applications of AF to other challenging unsolved problems such as the
zero-equals-one conjecture (which is also true) and the side-counting problem
of the circle. We determine the circle to have exactly 11 unique sides.

Proof of the Riemann Hypothesis utilizing the

theory of Alternative Facts
Donald J. Trump
January 24, 2017
Abstract
Conway’s powerful theory of Alternative Facts can render many difficult problems tractable. Here we demonstrate the power of AF to prove
the Riemann Hypothesis, one of the most important unsolved problems in
mathematics. We further suggest applications of AF to other challenging
unsolved problems such as the zero-equals-one conjecture (which is also
true) and the side-counting problem of the circle. We determine the circle
to have exactly 11 unique sides.

1

Problem statement

The Riemann Hypothesis has remained one of the most important unsolved
problems in modern mathematics. RH connects discrete mathematics to analytic mathematics and its truth or falsity has implications for hundreds of other
important unsolved questions from number theory to topology to algebra. RH
has until now resisted solution under mathematical theories based on standard
facts.
The Riemann Hypothesis states that the zeros of Riemann ζ function occur
only at the negative even integers and points in the complex plane with real
part equal to 1/2. We’ll apply Conway’s Alternative Fact framework to prove
this conjecture to be true.

2

Conway’s Alternative Facts

Conway’s Theory of Alternative Facts (AF) is a recent break-through in mathematical logic allowing for propositions to be either True or False depending on
context or motive of the agent positing the fact. While the traditional logic,
first introduced by Aristotle and formalized in the 19th century, associates a
unique Boolean value to any proposition, AF allows for more fluid associations. For any proposition, P , we associate a context C and a motive M and
a decision function which maps the triplet (P, C, M ) to the space of Booleans,
f : (P, C, M ) → {0, 1}.

1

The breakthrough with Alternative Facts is the concept of the context and
motive. A context is a maximal set of propositons, axiomatic or otherwise,
which can simultaneously be considered true. While the context as such does
not offer anything beyond the standard concept of formal systems, they become
considerably more powerful when we combine them with motives. A motive,
intuitively speaking, describe, the purpose of the propositons and the reasoning
steps that can be applied to them. Formally, they consist of constraints on the
lengths of logical chains of reasoning within the system. They separate logical
chains of reasoning into pro-motive chains and counter-motive chains. Promotive chains are the logical chains of reasoning that lead from propositions
explicitly included in the context to the desired logical conclusion. Countermotive chains lead from the same propositons to the negation of the desired
conclusion. Any chain of reasoning has a length consisting of the number of
logical steps taken. Within a motive, we define limits to each class of chain.
Typically, the length limit of counter-motive chains is chosen to be smaller than
for pro-motive chains.
A system of Alternative Facts is considered consistent when any attempt
to establish a contradiction within the system requires more logical steps than
allowed by the length limit of counter-motive chains. This is particularly useful
in using AFs to arrive at a desired outcome when reasoning time and space
is limited such as often occurs in situations such as class room lectures, print
publication and television interviews.

3

Proof of the Riemann Hypothesis using AF

We first posit the proposition which is the standard statement of the RH. In
this case, we can take the context to be a private one consisting solely of this
proposition and the statement that the ζ function is equal to the function θ(z) =
z − 1/2 at all points in the positive real half-plane and analytically continued
outside of that plane. The motive, M , is taken simply as the desire to prove
RH and we choose a limit of ∞ and 0 for pro-motive and counter-motive chains
respectively.
This function clearly has a unique critical point at z = 1/2 therefore proving
the RH within this context and under this motive. By extending this context
to include all other mathematical statements, except those contradicting RH,
we prove RH in the most general context in which it is true. Maximally general
contexts are referred to as huge.
We note however that this does not preclude the existence of other contexts
and motives in which the hypothesis is false. We leave as an open problem the
question of whether such a context exists and whether it is huge.

2

4

Difficulties involved in standard proofs

Here we point out the critical part played by AF in this proof. In a traditional
fact framework, positing the equality of ζ(z) and our θ(z) runs into several
problems. By utilizing the mechanism of Selberg’s trace formula or alternatively
Mangold’s explicit formula one can show that every prime number is equal to
either 4 or 8 in contradiction with the known factorizations of 4 and 8 and the
known impossibility of factoring 3, 5 and other larger known prime numbers.
Attempts to find a way around these difficulties utilizing psuedo-Riemannian
manifolds, p-adic circular abrigation and quasi-formal induction have all proved
dead ends.
These failures to prove RH by equating ζ with the θ-function all fail for
the same reason. In each case, it leads to the existence of propositions and
their negation both of which can be proven true. The existence of propositions
that are both true and false is impossible in the standard Aristotelian logic.
And so, progress on the problem was blocked until Conways’s breakthrough in
introducing Alternative Fact ontologies.

5

The Zero-Equals-One conjecture

Mathematicians have long wondered whether a consistent formal system can be
constructed in which the normal rules of arithmetic hold and in addition 0 = 1.
The chief difficulty in these model concerns the construction of higher numbers
when adding 1 must result in the same value as adding 0, which through the
systems axioms, must leave the number unchanged. If the original number must
be 0 or 1 (which by construction are identical anyway), forming numbers not
equal to these is challenging. We will now show how Conways AF theory can
provide a way forward.
We choose as our context the standard Peano Axoims for the set of natural
numbers and the identity 0=1. Our motive is the construction of higher numbers
and, as above, we limit the number of counter-motive steps to a small number.
A length of 1 suffices. We then construct the higher numbers by adding the
number 1. Since by the axioms, adding 1 results in a different number, we
achieve our result.
Now adding 1 to a number must be the same as adding 0 to a number and,
through our axioms, adding 0 to a number results in the same number. This is
therefore a contradiction. However this chain required utilizing two base axioms
and the application of 2 logical steps which lies outside of our stated motive.

6

Circles have 11 sides

The proof of this statement is straight-forward. Context is that circles have 11
sides and that all other properties of circles from Euclidean geometry hold. This,
therefore is a huge context. Motive is simply to avoid the existence of circles

3

with a number of sides unequal to 11. We will prove this using the technique of
reductio ad absurdum.
Assume that a circle of radius R, with R > 1, has N sides. Since N is a finite
number we can compare the number of sides to the known number of 11 using
the side-counting procedure. This requires N logical steps and so as long as we
allow for pro-motive logical chain lengths of at least N , we can construct such
a chain for any N . If N is unequal to 11 for any N (other than 11 of course),
we have arrived at a contradiction within out motive which proves that circles
must have exactly 11 sides.
Finally, our motive must limit counter-motive chains lengths to 10 which
precludes the existence of logical arguments based on side-counting that would
conflict with other known results of plane geometry that depend on circles not
having a finite number of sides. This completes the proof. The case of R ≤ 1 is
nearly identical and so we leave this to the reader.

7

Conclusion

Using Conways’s theory of Alternative Facts we prove the famous Riemann
Hypothesis. Our proof is constructive, providing an explicit context and motive
in which the theorem holds. In addition, we show how the power of AF can
be used to solve other challenging unsolved problems. We expect the methods
employed herein to generalize to many other unsolved problems and even some
proven theorems that can now also be proven otherwise.
Alternative facts have been incorrectly compared to falsehoods or out-right
lies. We hope in this paper to have corrected that misperception. Moreover,
some have argued that utilizing alternative facts allows for anything to be declared true making argumentation itself pointless. But this is not the case. As
long as two parties agree on a context and motive, any argumentation becomes
just as binding as an argument with standard facts. Difficulties may indeed arise
when context and motive are different. However, even in that case, the only
limitation is the inability of the two parties to reach a unique set of conclusions.
It is our experience that arguments between parties with different context and
motives and expectations of agreement rarely occur in practice.